The eta-inverted R-motivic sphere

TL;DR

This paper computes the eta-inverted R-motivic stable homotopy groups using spectral sequences, revealing a structure similar to the classical image of J and analyzing Toda brackets.

Contribution

It introduces a method combining Bockstein and Adams spectral sequences to calculate eta-inverted R-motivic homotopy groups, providing new structural insights.

Findings

Milnor-Witt (4k-1)-stem has order 2^{u+1}

Structure resembles classical image of J

Analyzes Toda bracket structure of the groups

Abstract

We use an Adams spectral sequence to calculate the R-motivic stable homotopy groups after inverting eta. The first step is to apply a Bockstein spectral sequence in order to obtain h_1-inverted R-motivic Ext groups, which serve as the input to the eta-inverted R-motivic Adams spectral sequence. The second step is to analyze Adams differentials. The final answer is that the Milnor-Witt (4k-1)-stem has order 2^{u+1}, where u is the 2-adic valuation of 4k. This answer is reminiscent of the classical image of J. We also explore some of the Toda bracket structure of the eta-inverted R-motivic stable homotopy groups.